Singularity Chaser wrote:
>
> Use Mathematica (or other methods) to find the asymptotic expansion of
> the following Sum in the 0+ limit
>
> Ksi(x)=Sum[1/n^(n*x),{n,1,Infinity}]
> [...]
Asymptotically it behaves as Exp[ProductLog[1/x]. More specifically, it
can be written as
(1+f[x])*Exp[ProductLog[1/x]]
where f[x]->0 as x->0.
I think f[x] is (at least eventually) negative, but am unable to pin it
down closer than that. To give an idea of the accuracy of this
approximation, for x=1/10000 we have a sum around 1326 whereas
Exp[ProductLog[10000]] is around 1383.
It would be useful to know why this limiting behavior is of interest.
Daniel Lichtblau
Wolfram Research